1. Field of the Invention
The invention relates to a method and apparatus for determining a malfunction of a technical system that is subject to a malfunction.
2. Description of the Related Art
The numerical simulation of electrical circuits has become of great significance in the development of computer chips in recent years. Simulators have become indispensable due to the high costs of producing a specimen chip and of possible redesign. With simulators it is possible to obtain predictive statements about operating behavior and efficiency of the modeled circuit on a computer. After successful simulation results, a chip is usually burned in silicon.
Using a xe2x80x9cNetworkxe2x80x9d approach, a circuit is described by its topological properties, the characteristic equations of circuit elements, and the Kirchhoff rules.
A modified, node analysis known from A. F. Schwarz, Computer-Aided design of microelectronic circuits and systems, vol. 1, Academic Press, London, ISBN 0-12-632431-A, pp. 185-188, 1987 (xe2x80x9cSchwarzxe2x80x9d) may be utilized for the analysis of a circuit. This leads to a differential-algebraic equation system having the form
C(x(t))xc2x7x(t)+f(x(t))+s(t)=0.xe2x80x83xe2x80x83(1)
A differential-algebraic equation system is an equation system of the type
F(x(t), x(t)t)=0xe2x80x83xe2x80x83(2)
with a singular Jacobi matrix Fxc2x7x of the partial derivations of F according to x(t).
The differential-algebraic equation system (1) is also called a quasi linear-implicit equation system. x(t) indicates a curve of node voltages dependent on a time t, x(t) indicates its derivative according to the time t. f(x(t)) references a first predetermined function that contains conductance values and non-linear elements, C(x(t)) references a predetermined capacitance matrix and s(t) references a second predetermined function that contains independent voltage sources and current sources.
Equation (1) describes only the ideal case for a circuit. In practice, however, noise, i.e., a malfunction of the circuit, cannot be avoided. xe2x80x9cNoisexe2x80x9d is defined here as an unwanted signal disturbance that is caused, for example, by thermal effects or the discrete structure of the elementary charge. Due to the increasing integration density of integrated circuits, the significance of the predictive analysis of such effects (noise simulation) is increasing.
In the analysis of a circuit taking noise into consideration, Equation (1) can be modeled as:
C(x(t))xc2x7x(t)+f(x(t))+s(t)+B(t, x(t))xc2x7xcexd(xcfx89, t)=0.xe2x80x83xe2x80x83(3)
where xcexd(xcfx89, t), references an m-dimensional vector whose independent components are generalized white noise, where m indicates the number of noise current sources. A matrix B(t, x(t), also referred to as intensity matrix has the dimension nxc3x97m. When only thermal noise in linear resistors occurs, then it is constant.
When the circuit is modelled with purely linear elements, then rule (3) becomes a disturbed, linear-implicit differential-algebraic equation system having the form
Cxc2x7x(t)+Gxc2x7x(t)+s(t)+B(t, x(t))xc2x7xcexd(xcfx89, t)=0.xe2x80x83xe2x80x83(4)
What is to be understood by the term xe2x80x9cindexxe2x80x9d below is a criterion regarding how far a differential-algebraic equation system xe2x80x9cdiffersxe2x80x9d from an explicit, ordinary differential equation system, how many derivation steps are required in order to obtain an explicit, ordinary differential equation system from the differential-algebraic equation system.
Without limitation of the universal validity and given existence of various term definitions of the term xe2x80x9cindexxe2x80x9d, the following definition is employed below for an index of a differential-algebraic equation system:
Let a differential-algebraic equation system of the type
F(x(t), x(t), t)=0xe2x80x83xe2x80x83(2)
be given. When a lowest natural number i exists, so that the equations
F(x(t), x(t), t)=0,xe2x80x83xe2x80x83(5)
                                                        ⅆ                              F                ⁡                                  (                                                                                    x                        .                                            ⁡                                              (                        t                        )                                                              ,                                          x                      ⁡                                              (                        t                        )                                                              ,                    t                                    )                                                                    ⅆ              t                                =          0                ,                            (        6        )                                          xe2x80x83                ⁢                  ⋮          ⁢                      xe2x80x83                    ,                                    xe2x80x83                                                      ⅆ                                                                                                   (                    i                    )                                                  ⁢                F                            ⁡                              (                                                                            x                      .                                        ⁡                                          (                      t                      )                                                        ,                                      x                    ⁡                                          (                      t                      )                                                        ,                  t                                )                                                          ⅆ                          t                              (                i                )                                                    =        0                            (        7        )            
can be transformed into a system of explicit ordinary differential equations, then xe2x80x98ixe2x80x99 is referred to as the index of the differential-algebraic equation system. The function F is thereby assumed to be capable of being differentiated an adequate number of times.
A xe2x80x9cstochastic differential equation systemxe2x80x9d is defined by the following differential equation system:
Let a Wiener-Hopf process {Wt; txc2x7xcex5R0+} be given on a probability space (xcexa9, A, P) together with a canonic filtration {Cs; sxcex5[a, b]}. Additionally h and G be: [a, b]xc3x97Rxe2x86x92R two ((B[a,b]xc3x97B)xe2x88x92B)-measurable random variables and {tilde over (X)}:xcexa9xe2x86x92R a (Caxe2x88x92B)-measurable function. A stochastic differential equation system is established by the Itxc3x4 differential
xe2x80x83Xs: xcexa9xe2x86x92R,
                    ω        ↦                                            X              ~                        ⁡                          (              ω              )                                +                                    ∫                              [                                  a                  ,                  s                                ]                                      ⁢                                          h                ⁡                                  (                                      u                    ,                                                                  X                        u                                            ⁡                                              (                        ω                        )                                                                              )                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  λ                  ⁡                                      (                    u                    )                                                                                +                                    (                                                ∫                  a                  s                                ⁢                                                      G                    ⁡                                          (                                              u                        ,                                                  X                          u                                                                    )                                                        ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                                          W                      u                                                                                  )                        ⁢                          (              ω              )                                                          (        8        )            
or, symbolically,
dXs=h(s, Xs)ds+G(s, Xs)dWs, sxcex5[a, b]; Xa={tilde over (X)}.xe2x80x83xe2x80x83(9)
The following method for noise simulation is known from A. Demir et al. Time-domain non-Monte Carlo noise simulation for nonlinear dynamic circuits with arbitrary excitations, IEEE Transactions on Computer-Aided Design of Integrated Circuits and System, Vol. 15, No. 5, pp.493-505, May 1996 (xe2x80x9cDemirxe2x80x9d)
For the case of a purely additive disturbance, rule (4) can be decoupled into a differential and an algebraic part.
The following applies in the case of a purely additive disturbance:
B(t, x(t))xe2x89xa1B(t),xe2x80x83xe2x80x83(10)
i.e., the intensity matrix is only dependent on the time t.
(4) thus becomes
Cxc2x7x(t)+Gxc2x7x(t)+s(t)+B(t)xc2x7xcexd(xcfx89, t)=0.xe2x80x83xe2x80x83(11)
Given the assumptions that consistent starting values (xdet(t0), xdet(t0)) are established at a starting time to and given regularity of the matrix brush {xcexC+G; xcexxcex5C}, an unambiguous solution xdet for (1) exists in the form
Cxc2x7xdet+Gxc2x7xdet+s=0.xe2x80x83xe2x80x83(12)
A matrix brush {xcexC+G; xcexxcex5C} is regular when a xcex0 from C exists such that
det(xcex0C+G)xe2x89xa00xe2x80x83xe2x80x83(13)
applies.
Consistent starting values (xdet(t0), xdet(t0) can be acquired in that a DC operating point of the system, which is described by (12), is defined, i.e., xdet=0 is set. NB. J. Leimkuhler et al., Approximation methods for the consistent initialization of differential-algebraic systems of equations, SIAM J. Numer. Anal., Vol. 28, pp. 205-226, 1991 (xe2x80x9cLeimkuhlerxe2x80x9d) discloses a further method for determining consistent starting values (xdet(t0), xdet(t0)).
After a transformation, which is described below, one arrives at rules that are equivalent to rule(11) and have the following form:
yxc2x7[1]+F1xc2x7y[1]+F2xc2x7y[2]+"sgr"[1]+({tilde over (B)}(t)xc2x7xcexd(xcfx89, t))[1]=0xe2x80x83xe2x80x83(14)
and
F3xc2x7y[1]+F4xc2x7y[2]+"sgr"[2]+({tilde over (B)}(t)xc2x7xcexd(xcfx89, t))[2]=0xe2x80x83xe2x80x83(15)
with transformed starting conditions
xe2x80x83y[1]det(t0):=(Txe2x88x921xc2x7xdet(t0))[1],xe2x80x83xe2x80x83(16)
y[1]det(t0):=(Txe2x88x921xc2x7xdet(t0))[1]xe2x80x83xe2x80x83(17)
and
y[2]det(t0):=(Txe2x88x921xc2x7xdet(t0))[2],xe2x80x83xe2x80x83(18)
y[2]det(t0):=(Txe2x88x921xc2x7xdet(t0))[2]xe2x80x83xe2x80x83(19)
and with
{tilde over (B)}(t):=Sxc2x7B(t).xe2x80x83xe2x80x83(20)
A prescribable matrix is respectively referenced with Fi, i=1, 2, 3, 4.
The goal of the transformation is to convert the rule (11) into a semi-explicit differential equation system of the type of rules (14) and (15) in the variable y=(y[1], y[2])T with suitable matrices Fi and a functions "sgr"=("sgr"[1], "sgr"[2])T.
Two regular matrices S and T are found thereto for matrix C from (11) that the rule
Sxc2x7Cxc2x7T=blockdiag(I, N)xe2x80x83xe2x80x83(21)
is satisfied, where I references a unit matrix having the dimension r, and N references a zero matrix having the dimension (nxe2x88x92r).
On the basis of Gauss elimination with complete pivot strategy, two regular matrices P1 and Q1 are determined with
P1xc2x7Cxc2x7Q1=C1,xe2x80x83xe2x80x83(22)
where a matrix C1 is a right upper triangle matrix whose r first diagonal elements are unequal to the value zero. Beginning from the (r+1)st line inclusive, the matrix C1 has only entries with the value zero. The matrix Q1 is selected as an orthogonal column-permutation matrix. The matrix P1 is the product of a lower left triangle matrix and an orthogonal row-permutation matrix.
Due to a multiplication by a regular upper right triangle matrix M1 from the right, all entries of the matrix C1 above its diagonals are eliminated:                               C          2                :=                                            C              1                        ·                          M              1                                =                      diag            ⁢                          "AutoLeftMatch"                              xe2x80x83                            ⁢                                                (                                                            λ                      1                                        ,                    …                    ⁢                                          xe2x80x83                                        ,                                          λ                      m                                        ,                                          xe2x80x83                                        ⁢                                                                  0                        ,                        …                        ⁢                                                  xe2x80x83                                                ,                        0                                                                    ⏟                                                                              (                                                          n                              -                              r                                                        )                                                                                -                            times                                                                                                                                )                                .                                                                        (        23        )            
Due to a multiplication by a regular diagonal matrix M2 from the left, all non-disappearing diagonal elements of the matrix C2 are transformed to the value 1:                               C          3                :=                                            M              2                        ·                          C              2                                =                      diag            ⁢                          "AutoLeftMatch"                              xe2x80x83                            ⁢                                                (                                                                                    1                        ,                        …                        ⁢                                                  xe2x80x83                                                ,                        1                                                                    ⏟                                                  r                                                      -                            times                                                                                                                ⁢                                          xe2x80x83                                        ,                                          xe2x80x83                                        ⁢                                                                  0                        ,                        …                        ⁢                                                  xe2x80x83                                                ,                        0                                                                    ⏟                                                                              (                                                          n                              -                              r                                                        )                                                                                -                            times                                                                                                                                )                                .                                                                        (        24        )            
The matrices S:=M2xc2x7P1 and T:=Q1xc2x7M1 perform, the necessary function.
By setting y:=Txe2x88x921xc2x7x, E:=C3=Sxc2x7Cxc2x7T, F:=Sxc2x7Gxc2x7T and "sgr"=Sxc2x7sin rule (11),
Exc2x7y+Fxc2x7y+"sgr"+{tilde over (B)}xc2x7xcexd=0xe2x80x83xe2x80x83(25)
derives.
In order to exploit the specific structure of the matrix C3, y is divided into a first vector y[1] that contains the first r components and into a second vector y[2] that contains the remaining (nxe2x88x92r) entries:
y=(y[1], y[2])T.xe2x80x83xe2x80x83(26)
The matrix F is divided into 4 sub-matrices Fi, i=1, 2, 3, 4 having the dimensions rxc3x97r, rxc3x97(nxe2x88x92r), (nxe2x88x92r)xc3x97r, (nxe2x88x92r)xc3x97(nxe2x88x92r):                     F        =                              (                                                            F                  1                                |                                  F                  2                                                                              F                  3                                |                                  F                  4                                                      )                    .                                    (        27        )            
A corresponding division is selected for the matrix E.
The matrix E1 is a unit matrix having the dimension rxc3x97r, and the matrices E2, E3 and E4 are zero matrices. (25) thus resolves into the two rules (14) and (15).
When the matrix can be inverted, which is precisely the case when the system from rule (12) has the index 1, then rule (15),
F3xc2x7y[1]+F4xc2x7y[2]+"sgr"[2]+({tilde over (B)}(t)xc2x7xcexd(xcfx89, t))[2]=0xe2x80x83xe2x80x83(15)
can be solved for y[2], which leads to the following rule:
y[2]=xe2x88x92F4xe2x88x921xc2x7{F3xc2x7y[1]+"sgr"[2]+({tilde over (B)}(t)xc2x7xcexd(xcfx89, t))[2]}.xe2x80x83xe2x80x83(28)
The following, abbreviating terms are introduced below:
{circumflex over (F)}:=F1xe2x88x92F2xc2x7F4xe2x88x921xc2x7F3;xe2x80x83xe2x80x83(29)
Ŝ:="sgr"[1]xe2x88x92F2xc2x7F4xe2x88x921xc2x7"sgr"[2];xe2x80x83xe2x80x83(30)
{circumflex over (B)}(t, x):=(Ir, xe2x88x92F2xc2x7F4xe2x88x921)Txc2x7{circumflex over (B)}(t).xe2x80x83xe2x80x83(31)
Ir references a unit matrix having the dimension r, i.e., the rank of the matrix C.
Inserting rule (28) into rule (14), produces:
yxc2x7[1]+{circumflex over (F)}xc2x7y[1]Ŝ+{circumflex over (B)}(t)xc2x7xcexd(xcfx89, t)=0.xe2x80x83xe2x80x83(32)
Rule (32) can be interpreted as a stochastic differential equation system having the following form:                               ⅆ                      Y            t                          [              1              ]                                      =                              Y                          t              0                                      [              1              ]                                -                                    {                                                F                  ^                                ·                                  Y                  t                                      [                    1                    ]                                                  ·                                  s                  ^                                            }                        ⁢                          ⅆ              t                                -                                                    B                ^                            ⁡                              (                t                )                                      ⁢                                          ⅆ                                  W                  t                                            .                                                          (        33        )            
A random variable with an anticipation value       y    det          [      1      ]        ⁢      (          t      0        )  
that exhibits a finite variance is reference       Y          t      0              [      1      ]        ·      {                  W        t            ;              t        ∈                  ℜ          0          +                      }  
is a Wiener-Hopf process having the dimension of the number of noise sources, generally the number of disturbing sources.
The method from [2] proceeds from rule (33) whose unambiguous solution process Yt[1] is established as Itxc3x4 differential by the equation                               Y          T                      [            1            ]                          =                                            Φ                              t                ,                                  t                  0                                                      ·                          Y                              t                0                                            [                1                ]                                              -                      (                                                            ∫                                      [                                                                  t                        0                                            ,                      t                                        ]                                                  ⁢                                                                            Φ                                              t                        ,                        u                                                              ·                                                                  s                        ^                                            ⁡                                              (                        u                        )                                                                              ⁢                                      ⅆ                                          λ                      ⁡                                              (                        u                        )                                                                                                        +                                                ∫                                      t                    0                                    t                                ⁢                                                                            Φ                                              t                        ,                        u                                                              ·                                                                  B                        ^                                            ⁡                                              (                        u                        )                                                                              ⁢                                      ⅆ                                          W                      u                                                                                            )                                              (        34        )            
with the fundamental system of solutions                               Φ                      t            ,                          t              0                                      =                              exp            ⁡                          (                              -                                                      ∫                                          [                                                                        t                          0                                                ,                        t                                            ]                                                        ⁢                                                            F                      ^                                        ⁢                                          ⅆ                                              λ                        ⁡                                                  (                          u                          )                                                                                                                                )                                =                      exp            ⁢                          {                                                (                                                            t                      0                                        -                    t                                    )                                ⁢                                  F                  ^                                            }                                                          (        35        )            
Given the method from [2], the anticipation values Et and the second moments Pt of the random variables Yt[1] are approximately determined. The anticipation value of an Itxc3x4 integral is equal to the value 0. Thus,                                           E            t                    ⁢                      :                          =                              E            ⁡                          (                              Y                t                                  [                  1                  ]                                            )                                =                                                    Φ                                  t                  ,                                      t                    0                                                              ·                                                y                  det                                      [                    1                    ]                                                  ⁡                                  (                                      t                    0                                    )                                                      -                                          ∫                                  [                                                            t                      0                                        ,                    t                                    ]                                            ⁢                                                                    Φ                                          t                      ,                      u                                                        ·                                                            s                      ^                                        ⁡                                          (                      u                      )                                                                      ⁢                                  ⅆ                                      λ                    ⁡                                          (                      u                      )                                                                                                                              (        35        )            
is directly obtained from rule (34)
For all t, Et solves the ordinary differential equation system
                              x                      t            0                          =                                            y              det                              [                1                ]                                      ⁡                          (                              t                0                            )                                .                                    (        36        )            
For all t, the second moments                               P          t                =                  E          ⁢                      {                                          (                                  Y                  t                                      [                    1                    ]                                                  )                            2                        }                                              (        37        )            
of the random variables Yt[1] of the solution process satisfy the differential equation
{fraction (Pt/dt)}=xe2x88x92Fxc2x7Ptxe2x88x92Ptxc2x7FTxe2x88x92ŝxc2x7EtTxe2x88x92Etxc2x7ŝT+{circumflex over (B)}(t)xc2x7(t)T,xe2x80x83xe2x80x83(38)
where a starting condition is established by                               P                      t            0                          =                  E          ⁢                      {                                          Y                                  t                  0                                                  [                  1                  ]                                            ·                                                (                                      Y                                          t                      0                                                              [                      1                      ]                                                        )                                T                                      }                                              (        39        )            
(38) is a matter of a linear ordinary differential equation system.
Parallel to the transient simulation of the circuit, the anticipation values Et and the second moments Pt are approximately determined in the method from [2] by numerical integration with linear, implicit multi-step methods.
One disadvantage of this method is that linear equation systems of the quadratic order must be solved in the plurality m of the noise sources for each time step for determining the second moments Pt.
This method is based on a manual index reduction of the differential-algebraic equation system to an explicit, stochastic differential equation system that cannot be automated. Moreover, only some of the noise effects are considered. Furthermore, this method does not supply any path-wise information but only the moments of the node potentials that are preserved in the index reduction. This method supplies no information for the node potentials that are suppressed by the index reduction.
A further disadvantage of the method found in Demir is that the index reduction in this method is extremely inefficient and can also not be automatically implemented, particularly since the algebraic variables are not taken into consideration in the differential-algebraic equation system. Given the method from Demir, the index reduction must be analytically manually carried out since the numerical methods are not stable.
It is also known from L. O. Chua and P. M. Lin, Computer aided design of electronic circuits, Prentice Hall, Englewood Cliffs, 1975, ISBN 0-13-165415-2, p. 596 (xe2x80x9cChuaxe2x80x9d) to implement a noise simulation of an integrated circuit in the frequency domain, but this results in a circuit that can only be analyzed in the small signal area and the prerequisite of a fixed operating point, is frequently not established. For example, the resonant behavior of an oscillator in a circuit prevents a uniform operating point. W. Nagel, SPICE2xe2x80x94a computer program to simulate semiconductor circuits, Tech. Report, UC Berkeley, Memo ERL-M 520, 1975 (xe2x80x9cNagelxe2x80x9d) circuit can be described in a form that can be processed by a computer. P. E. Kloeden and E. Platen, Numerical (I solution of stochastic differential equations, Springer Verlag, Berlin, N.Y., ISBN 3-540-54062-8, p.412, 1992 (xe2x80x9cKloedenxe2x80x9d) discloses a method for the numerical handling of a stochastic differential equation system, the path-wise simulation of discrete approximations in the solution process, that is referred to as the Runge-Kutta strategy.
U.S. Pat. No. 5,646,869 discloses a simulator that comprises
an initialization unit,
an incrementing unit,
a unit for updating an estimated value, and
an output unit.
An initialization value of a status and of a function with which a system that is subject to a random disturbance is described; this value is supplied from the initialization unit to the incrementing unit. The incrementing unit employs two random number sequences in order to form an increment of the estimated value without differentiating the function itself. The estimated value is incremented by the increment in the unit for updating an estimated value.
Furthermore, U.S. Pat. No. 5,132,897 discloses a method as well as an apparatus for improving the precision of a system controlled in a closed control circuit, in which at least two stochastic noise signals are taken into consideration.
The invention is thus based on providing a method which avoids the above disadvantages.
For the inventive method, a technical system that is subject to a malfunction is described with an implicit stochastic differential equation system (SDE) exhibiting the form C*x(t)+G*x(t)+s(t)+B*xcexd(t)=0, where C references a first matrix, x(t) references a derivation of a status of the system after a time that is described by a time variable t, G references a second matrix, s(t) references a system function of independent system quantities, B references a third matrix and v(t) references a noise function. An approximate solution of this system is determined in that a discrete approximation process is realized. The discrete approximation process is realized according to the following rule:
xe2x88x92(C+hxcex12G)xc2x7{tilde over (X)}xcfx80n+1={xe2x88x92(1xe2x88x92xcex3)C+h(1+xcex3xcex11xe2x88x92xcex12)G}xc2x7{tilde over (X)}xcfx80n+
+{xe2x88x92xcex3C+h(xcex3xe2x88x92xcex11xcex3)G}xc2x7{tilde over (X)}xcfx80n+1
+hxcex12s(xcfx80n+1)+h(1+xcex3xcex11xe2x88x92xcex12)s(xcfx80n)+
+h(xcex3xe2x88x92xcex11xcex3)s(xcfx80nxe2x88x921)+
+B(xcfx80n)xc2x7xcex94Wn+xcex3B(xcfx80n)xc2x7xcex94Wnxe2x88x921
C is a first matrix
xcex11, xcex12, xcex3 are predetermined parameters from the interval [0, 1],             h      ⁢              :              =          T      N        ,
xe2x80x83is a step width in an output time interval [0, T], whereby T is a predetermined value that is subdivided into N sub-intervals,
G is a second matrix,
{tilde over (X)}xcfx80n+1 is a realization of the approximation process at a supporting point xcfx80n+1,
{tilde over (X)}xcfx80nxe2x88x921 is a realization of the approximation process at a supporting point xcfx80nxe2x88x921,
S(xcfx80n+1) is a realization of the approximation process at a supporting point xcfx80n+1,
S(xcfx80nxe2x88x921) is a first value at the supporting point xcfx80nxe2x88x921,
xcex94Wnxe2x88x921:=Wxcfx80nxe2x88x92Wxcfx80nxe2x88x921 is a difference value between a second value Wxcfx80n at the supporting point xcfx80n and the second value Wxcfx80nxe2x88x921 at the supporting point xcfx80nxe2x88x921,
B(xcfx80n) is a second value at the supporting point xcfx80n, and
where the disturbance {tilde over (X)}xcfx80n+1 is determined by an iterative solution of the approximation process.
The inventive apparatus comprises a processor unit that is configured such that a technical system that is subject to a malfunction is described with an implicit stochastic differential equation system of the form C*{dot over (x)}(t)+G*x(t)+s(t)+B*v(t)=0, where C references a first matrix, {dot over (x)}(t) references a derivation of a status of the system after a time that is described by a time variable t, G references a second matrix, s(t) references a system function of independent system quantities, B references a third matrix and v(t) references a noise function. An approximate solution of this system is determined in that a discrete approximation process is realized. The discrete approximation process is realized according to the following rule:
xe2x88x92(C+hxcex12G)xc2x7{tilde over (X)}xcfx80n+1={xe2x88x92(1xe2x88x92xcex3)c+h(1+xcex3xcex11xe2x88x92xcex12)G}xc2x7{tilde over (X)}xcfx80nxe2x88x921+
+{xe2x88x92xcex3C+h(xcex3xe2x88x92xcex11xcex3)G}xc2x7{tilde over (X)}xcfx80n+1
+hxcex12s(xcfx80n+1)+h(1+xcex3xcex11xe2x88x92xcex12)s(xcfx80n)+
+h(xcex3xe2x88x92xcex11xcex3)s(xcfx80nxe2x88x921)+
+B(xcfx80n)xc2x7xcex94Wn+xcex3B(xcfx80n)xc2x7xcex94Wnxe2x88x921
where the following connotations apply:
C is a first matrix
xcex11, xcex12, xcex3 are predetermined parameters from the interval [0, 1],             h      ⁢              :              =          T      N        ,
xe2x80x83is a step width in an output time interval [0, T], whereby T is a predetermined value that is subdivided into N sub-intervals,
G is a second matrix,
{tilde over (X)}xcfx80n+1 is a realization of the approximation process at a supporting point xcfx80n+1,
{tilde over (X)}xcfx80n is a realization of the approximation process at a supporting point xcfx80n,
{tilde over (X)}xcfx80nxe2x88x921 is a realization of the approximation process at a supporting point xcfx80nxe2x88x921,
S(xcfx80nxe2x88x921) is a first value at the supporting point xcfx80n+1,
S(xcfx80n) is a first value at the supporting point xcfx80n,
S(xcfx80nxe2x88x921) is a first value at the supporting point xcfx80nxe2x88x921,
xcex94Wnxe2x88x921:=Wxcfx80nxe2x88x92Wxcfx80nxe2x88x921 is a difference value between a second value Wxcfx80n at the supporting point xcfx80n and the second value Wxcfx80nxe2x88x921 at the supporting point xcfx80nxe2x88x921,
B(xcfx80n) is a second value at the supporting point xcfx80n. The disturbance {tilde over (X)}xcfx80n+1 is determined by an iterative solution of the approximation process.
The invention directly employs the implicit structure of the technical system, represented by an implicit differential equation system. As a result of the invention, the determination of the malfunction is considerably accelerated, since the thin occupancy of the matrices C and G can be exploited. The numerically unstable and involved transformation of the differential equation system into the decoupled form is eliminated.
The invention makes it possible for the first time to also determine a malfunction given a singular matrix C.
Other advantageous embodiments include the inventive method (and an apparatus for implementing the method), where the malfunction is noise which the system is subjected to, or where the malfunction is purely additive. A method may also be implemented where the malfunction is determined path-wise. A further step of interpolating realizations of the approximation process may be used to determine a steady approximation process {{tilde over (X)}s; sxcex5[0, T]}. Identified paths determined path-wise may be analyzed with a statistical method. This method may be used where the system is an electrical circuit, a mechanical multi-member system, a physical system, a chemical system, or a physical-chemical system.
The invention can be utilized whenever a technical system is disturbed and can be described by a system of differential-algebraic equations.
For example, disturbances (noise) in an electrical circuit can be determined. The invention is also suitable for employment in a mechanical multi-member system or in a general physical system, a chemical system or a physical-chemical system as well whose respective modeling leads to a system of differential-algebraic equations.